For each of the following, find the number of shortest lattice paths from (5,5) to (11,11) which: a. pass through the point (7,6). b. avoid (do not pass through) the point (9, 10). c. either pass through (7,6) or (9, 10) (or both).

3. A rectangular room has dimensions 12x12x24. That is, the floor and ceiling and both the side walls are 12x24 and the two end walls are 12x12. In the room there are two ants, a male and a female. The male ant is on the floor at one of the corners. Now the female has positioned herself to be as far as possible from the male. That is, she has located herself at a point so that the male will take the longest possible time to get to her, given that he has to craw! along the walls, floor or ceiling of the room and will (of course) choose his path so that he gets to the female in the shortest possible time. • F The question is: where is the female? Well there's an obvious answer-the diametrically opposite corner. That's certainly the point which is farthest from the ant "as the crow flies." But an ant is not a crow.

please,don't provied handwriting solution.

A company makes two kinds of animal food, A and B, which contain two food supplements. It takes 2 pounds of the first supplement and one pound of the second to make a dozen cans of food A, and 4 poundsof the first supplement and 5 pounds of the second to make a dozen cans of food B. On a certain day 80 pounds of the first supplement and 70 pounds of the second are available.   Use the geometrical approach tofind the number of cans of each food should be made to maximize profit, if the profit on a dozen cans of food A is $3 and the profit on a dozen cans of food B is $10. Show every necessary steps.